#7630811

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 44 points:

(2,4)(6,9)(10,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,28)(23,33)(25,36)(26,35)(27,32)(29,34)(30,38)(31,40)(37,42)(39,43)(41,44);(5,6)(7,10)(8,9)(11,12)(13,15,20,17)(14,21,16,24)(18,25,32,29)(19,26,33,30)(22,34,27,36)(23,35,28,38)(31,39,42,41)(37,43,40,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,14)(15,21,17,24)(18,27,32,22)(19,28,33,23)(25,34,29,36)(26,35,30,38)(31,40,42,37)(39,43,41,44);(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,21,16,24)(18,25,32,29)(19,30,33,26)(22,36,27,34)(23,35,28,38)(31,39,42,41)(37,44,40,43);(13,18)(14,22)(15,25)(16,27)(17,29)(19,31)(20,32)(21,34)(23,37)(24,36)(26,39)(28,40)(30,41)(33,42)(35,43)(38,44);(13,19)(14,23)(15,26)(16,28)(17,30)(18,31)(20,33)(21,35)(22,37)(24,38)(25,39)(27,40)(29,41)(32,42)(34,43)(36,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538007);